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1、ONLI{YORKEPAIRSFrancoisBlanchard,EliGlasner,SergiKolyadaandAlejandroMaassAbstract.TheLi{Yorkede nitionofchaosproveditsvalueforintervalmaps.Inthispaperitisconsideredinthesettingofgeneraltopologicaldynamics.Weadopttwooppositepointsofview.Ontheonehan
2、dsucientconditionsforLi{Yorkechaosinatopologicaldynamicalsystemaregiven.Wesolvealong{standingopenquestionbyprovingthatpositiveentropyimpliesLi{Yorkechaos.OntheotherhandpropertiesofdynamicalsystemswithoutLi{Yorkepairsareinvestigated;inadditiontohavin
3、gentropy0,theyareminimalwhentransitive,andthepropertyisstableunderfactormaps,arbitraryproductsandinverselimits.FinallyitisproventhatminimalsystemswithoutLi{Yorkepairsaredisjointfromscatteringsystems.0.IntroductionTheterm`chaos'inconnectionwithamapwas
4、introducedbyLiandYorke[LY],althoughwithoutaformalde nition.Todaytherearevariousde nitionsofwhatitmeansforamaptobechaotic,someofthemworkingreasonablyonlyinparticularphasespaces;mostoftheexistingoneswerereviewedin[KS].Althoughonecouldsaythat`asmanyauth
5、ors,asmanyde nitionsofchaos',mostofthemarebasedontheideaofunpredictabilityofthebehavioroftrajectorieswhenthepositionofthepointisgivenwithanerror(instabilityoftrajectoriesorsensitivedependenceoninitialconditionsaretermsusuallyusedtodescribethisphenome
6、non).Thepresentarticlemainlydealswithoneofthesede nitions,namelychaosinthesenseofLi{Yorke.Itisoneinaseriesofpapersbyvariousauthorsonthatsubject([Iw],[XY],[FDM]andtwoforthcomingpapers,[HY1],[HY2]).Ourmainpurposeistoincorporatethisnotionofchaosintotheg
7、eneralframeofTopologicalDynamics;formerlyithadbeenstudiedmainlyinthesettingofintervalmaps.Let(X;T)beatopologicaldynamicalsystem,withXacompactmetricspacewithmetric%,andTasurjectivecontinuousmapfromXtoitself.Thede nitionofLi{Yorkechaosisbasedonideasin[
8、LY].Apairofpointsfx;ygXissaidtobeaLi{Yorkepair(withmodulus)ifonehassimultaneouslylimsup%(Tnx;Tny)=>0andliminf%(Tnx;Tny)=0:n!1n!1AsetSXiscalledscrambledifanypairofdistinctpointsfx;ygSisaLi{Yorkepair.Finally,asystem(X;T)iscalledchaoticinthesenseof
